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Properties

Label 342.10.b.b
Level $342$
Weight $10$
Character orbit 342.b
Analytic conductor $176.142$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,10,Mod(341,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.341");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 342.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(176.142255968\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8} + 25536 q^{14} + 1966080 q^{16} - 323202 q^{19} - 5473830 q^{25} + 408576 q^{28} - 8475660 q^{29} + 31457280 q^{32} - 5171232 q^{38} + 20345076 q^{41} - 60729780 q^{43} + 214984890 q^{49} - 87581280 q^{50} - 181795212 q^{53} - 316864944 q^{55} + 6537216 q^{56} - 135610560 q^{58} + 197198784 q^{59} + 53410728 q^{61} + 503316480 q^{64} + 640935936 q^{65} - 335924712 q^{71} + 136407840 q^{73} - 82739712 q^{76} + 325521216 q^{82} + 269215776 q^{85} - 971676480 q^{86} - 1186853004 q^{89} + 506121804 q^{95} + 3439758240 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 16.0000 0 256.000 − 2419.62i 0 2482.82 4096.00 0 − 38713.9i
341.2 16.0000 0 256.000 − 2191.32i 0 5580.72 4096.00 0 − 35061.2i
341.3 16.0000 0 256.000 − 2036.36i 0 −10192.3 4096.00 0 − 32581.8i
341.4 16.0000 0 256.000 − 1991.51i 0 −6656.82 4096.00 0 − 31864.1i
341.5 16.0000 0 256.000 − 1660.24i 0 7882.01 4096.00 0 − 26563.8i
341.6 16.0000 0 256.000 − 1517.64i 0 −3027.22 4096.00 0 − 24282.3i
341.7 16.0000 0 256.000 − 1365.25i 0 −11786.9 4096.00 0 − 21844.0i
341.8 16.0000 0 256.000 − 1276.01i 0 −1734.40 4096.00 0 − 20416.1i
341.9 16.0000 0 256.000 − 1273.82i 0 10689.6 4096.00 0 − 20381.1i
341.10 16.0000 0 256.000 − 1184.89i 0 487.542 4096.00 0 − 18958.2i
341.11 16.0000 0 256.000 − 931.256i 0 3462.46 4096.00 0 − 14900.1i
341.12 16.0000 0 256.000 − 794.160i 0 1455.94 4096.00 0 − 12706.6i
341.13 16.0000 0 256.000 − 339.755i 0 −2928.32 4096.00 0 − 5436.09i
341.14 16.0000 0 256.000 − 207.823i 0 −6512.79 4096.00 0 − 3325.17i
341.15 16.0000 0 256.000 − 172.069i 0 11595.6 4096.00 0 − 2753.10i
341.16 16.0000 0 256.000 172.069i 0 11595.6 4096.00 0 2753.10i
341.17 16.0000 0 256.000 207.823i 0 −6512.79 4096.00 0 3325.17i
341.18 16.0000 0 256.000 339.755i 0 −2928.32 4096.00 0 5436.09i
341.19 16.0000 0 256.000 794.160i 0 1455.94 4096.00 0 12706.6i
341.20 16.0000 0 256.000 931.256i 0 3462.46 4096.00 0 14900.1i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.10.b.b yes 30
3.b odd 2 1 342.10.b.a ✓ 30
19.b odd 2 1 342.10.b.a ✓ 30
57.d even 2 1 inner 342.10.b.b yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.10.b.a ✓ 30 3.b odd 2 1
342.10.b.a ✓ 30 19.b odd 2 1
342.10.b.b yes 30 1.a even 1 1 trivial
342.10.b.b yes 30 57.d even 2 1 inner